L(s) = 1 | − 4·5-s + 18·13-s − 104·17-s − 109·25-s − 284·29-s + 214·37-s − 472·41-s − 343·49-s − 572·53-s + 830·61-s − 72·65-s − 1.09e3·73-s + 416·85-s + 176·89-s − 594·97-s + 1.94e3·101-s + 1.74e3·109-s + 1.32e3·113-s + ⋯ |
L(s) = 1 | − 0.357·5-s + 0.384·13-s − 1.48·17-s − 0.871·25-s − 1.81·29-s + 0.950·37-s − 1.79·41-s − 49-s − 1.48·53-s + 1.74·61-s − 0.137·65-s − 1.76·73-s + 0.530·85-s + 0.209·89-s − 0.621·97-s + 1.91·101-s + 1.53·109-s + 1.10·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 4 T + p^{3} T^{2} \) |
| 7 | \( 1 + p^{3} T^{2} \) |
| 11 | \( 1 + p^{3} T^{2} \) |
| 13 | \( 1 - 18 T + p^{3} T^{2} \) |
| 17 | \( 1 + 104 T + p^{3} T^{2} \) |
| 19 | \( 1 + p^{3} T^{2} \) |
| 23 | \( 1 + p^{3} T^{2} \) |
| 29 | \( 1 + 284 T + p^{3} T^{2} \) |
| 31 | \( 1 + p^{3} T^{2} \) |
| 37 | \( 1 - 214 T + p^{3} T^{2} \) |
| 41 | \( 1 + 472 T + p^{3} T^{2} \) |
| 43 | \( 1 + p^{3} T^{2} \) |
| 47 | \( 1 + p^{3} T^{2} \) |
| 53 | \( 1 + 572 T + p^{3} T^{2} \) |
| 59 | \( 1 + p^{3} T^{2} \) |
| 61 | \( 1 - 830 T + p^{3} T^{2} \) |
| 67 | \( 1 + p^{3} T^{2} \) |
| 71 | \( 1 + p^{3} T^{2} \) |
| 73 | \( 1 + 1098 T + p^{3} T^{2} \) |
| 79 | \( 1 + p^{3} T^{2} \) |
| 83 | \( 1 + p^{3} T^{2} \) |
| 89 | \( 1 - 176 T + p^{3} T^{2} \) |
| 97 | \( 1 + 594 T + p^{3} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.14404771264738954770926965809, −9.932835428434986362842795165433, −8.967773360671679158669725133312, −8.032255173369180197798135216655, −6.98088650850768157412595735817, −5.93115399905379624366103092437, −4.62001932386658590248533999721, −3.52469038851574480735221966530, −1.93125439260378926307959627855, 0,
1.93125439260378926307959627855, 3.52469038851574480735221966530, 4.62001932386658590248533999721, 5.93115399905379624366103092437, 6.98088650850768157412595735817, 8.032255173369180197798135216655, 8.967773360671679158669725133312, 9.932835428434986362842795165433, 11.14404771264738954770926965809